Unit V Big Questions
Where does the formula for the difference quotient come from? Well know i now this question because i learned it from the wonderful Mrs. Kirch :D
In this graph, we can see our understanding of where the difference quotient is derived from. The first point is (x, f(x)). There is delta x and delta equals change in distance between the first and second point meaning the second point is (x + delta x, f(x + delta x)) that is what they used delta x but, for this we will reference to delta x as 'h' so our second point can also be written as (x+h, f(x+h) so that we can get a better understanding of a variable that we know so it wont be so confusing. The line that touches the graph at two points is called the secant line, very different from a tangent line that only touches the graph once. To find the slope of the secant line we are going to use the slope formula. The slope formula is m = y2-y1 all divided by x2-x1. When we plug in our two points from the first graph we have f(x + delta x) - f(x) all over x + delta x - x. Then we substituted h for delta x and we get f(x+h) - f(x) all over x + h -x. Then we notice in the denominator we have a x and -x and what happens when you get a positive and negative number or variable is this case variable you CANCEL THEM, leaving only h. Then you get the Difference quotient f(x+h) - f(x) divided by the letter h, that's the difference quotient :D
There is a video that verbally shows you how to derive the difference quotient by clicking here
Works Cited
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